HOME    |    BROWSE    |    SEARCH    |    SUBSCRIPTION   |    SUBMISSION    |    REGISTRATION

Quick Search:
Author: Title/Abstract: Vol./No: Page:
PTP Online Top > Supplement (1976) > No.59
| Next Article >

Prog. Theor. Phys. Supplement No.59 (1976) pp. 1-35

[ Full Text PDF : FREE ACCESS (1982K) ]

Development of the Theory of a Nonlinear Lattice

Morikazu Toda

Institute for Applied Physics, Faculty of Engineering, Yokohama National University, Yokohama 233

(Received April 17, 1976)

Abstract:

An extensive account of the exact theory of wave propagation in the one-dimensional nonlinear lattice with exponential interaction between nearest neighbor particles is given. A brief review of the development, useful particular solutions, the general method of solving the equations of motion and the relation between the discrete lattice and the continuous Korteweg-de Vries system are given, with some future aspect of the problems of nonlinear lattices.


URL : http://ptp.ipap.jp/link?PTPS/59/1/
DOI : 10.1143/PTPS.59.1

[ Full Text PDF : FREE ACCESS (1982K) ] Citation:

References:

  1. E. Fermi, J. Pasta and S. Ulam, Collected Papers of Enrico Fermi, Vol. II (University of Chicago Press, 1965), p. 978.
  2. J. Ford, J. Math. Phys. 2 (1961), 387[CrossRef].
    J. Ford and J. Waters, J. Math. Phys. 4 (1963), 1293[CrossRef]; ibid. 7 (1966), 399[AIP Scitation].
  3. E. A. Jackson, J. Math. Phys. 4 (1963), 551[CrossRef].
  4. N. J. Zabusky and G. S. Deem, J. Comput. Phys. 2 (1968), 207.
    N. J. Zabusky, International Conference on Statistical Mechanics, Kyoto, 1968; J. Phys. Soc. Jpn. 26 (1969) Suppl., p. 196; Comput. Phys. Commun. 5 (1973), 1[CrossRef].
    See also Ref. 8) for the recurrence phenomenon.
  5. M. Toda, J. Phys. Soc. Jpn. 20 (1965), 2095A[JPSJ]; Prog. Theor. Phys. Suppl. No. 36 (1966), 113[PTP].
  6. M. Toda, J. Phys. Soc. Jpn. 22 (1967), 431[JPSJ].
  7. M. Toda, J. Phys. Soc. Jpn. 23 (1967), 501[JPSJ]; Physica Norvegica 5 (1971), 203.
  8. M. Toda, International Conference on Statistical Mechanics, Kyoto, 1968; J. Phys. Soc. Jpn. 26 (1969) Suppl., p. 235.
  9. M. Toda and M. Wadati, J. Phys. Soc. Jpn. 34 (1973), 18[JPSJ].
  10. R. Hirota, J. Phys. Soc. Jpn. 35 (1973), 286[JPSJ].
    See also Ref. 19) for N-soliton solution.
  11. M. Toda, Prog. Theor. Phys. Suppl. No. 45 (1970), 174[PTP].
  12. M. Toda, “Studies of a Non-linear Lattice”, Phys. Rep. C 18 (1975), 1, [CrossRef]which is nearly the same as M. Toda, Arkiv for Det Fysiske Seminar i Trondheim, No. 2-1974.
  13. M. Kac and P. van Moerbeke, J. Math. Phys. 72 (1974), 2879[AIP Scitation].
  14. E. Date and S. Tanaka, Prog. Theor. Phys. 55 (1976), 457[PTP]; Prog. Theor. Phys. Suppl. No. 59 (1976), 107[PTP].
  15. M. Toda and M. Wadati, J. Phys. Soc. Jpn. 39 (1975), 1204[JPSJ].
  16. M. Wadati and M. Toda, J. Phys. Soc. Jpn. 39 (1975), 1196[JPSJ].
  17. M. Toda, Lecture at the Conference on the Theory and Application of Solitons, Tucson, 1976.
  18. H. Chen and C. Liu, J. Math. Phys. 16 (1975), 1428[CrossRef].
  19. H. Flaschka, Phys. Rev. B 9 (1974), 1924[APS].
  20. M. Hénon, Phys. Rev. B 9 (1974), 1921[APS].
  21. H. Flaschka, Prog. Theor. Phys. 51 (1974), 703[PTP].
  22. D. W. McLaughlin, J. Math. Phys. 16 (1975), 96[CrossRef]; ibid. 16 (1975), 1704[CrossRef].
  23. H. Flaschka and McLaughlin, Prog. Theor. Phys. 55 (1976), 438[PTP].
  24. Dynamical Systems, Theory and Applications, Lecture Note in Physics 38 J. Moser, ed. (Springer Verlag, 1975), p. 467.
  25. N. J. Zabusky, Proceedings of the Symposium on Nonlinear Partial Differential Equations (Academic Press, 1967), p. 223.
  26. P. D. Lax, Commun. Pure Appl. Math. 21 (1968), 467.
  27. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 15 (1963), 240[APS].
  28. M. Toda, International Conference on Mathematical Problems in Physics, Kyoto, 1975, Lecture Notes in Physics 39 H. Araki, ed. (Springer Verlag, 1975), p. 387.
  29. D. J. Korteweg and G. de Vries, Phil. Mag. 18 (1895), 35.
  30. K. M .Case and M. Kac, J. Math. Phys. 14 (1973), 594[CrossRef].
  31. K. M. Case, J. Math. Phys. 14 (1973), 916[CrossRef].
  32. H. D. Wahlquist and F. B. Estabrook, Phys. Rev. Lett. 31 (1973), 1386[APS].
  33. M. Wadati, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975), 419[PTP].
  34. D. N. Payton III, R. Rich and W. M. Visscher, Phys. Rev. 160 (1967), 706[APS]; Phys. Rev. A 4 (1971), 1682[APS]; Proceedings of the International Conference on Localized Excitation in Solids, California, 1967 (Plenum Press, 1968), p. 657.
    E. A. Jackson, J. R. Pasta and J. F. Waters, J. Comput. Phys. 2 (1968), 1207.
  35. M. Toda, R. Hirota and J. Satsuma, Prog. Theor. Phys. Suppl. No. 59 (1976), 148[PTP].
  36. J. G. Berryman, from conversation.
  37. G. Casati and J. Ford, preprint.
  38. H. Ono, J. Phys. Soc. Jpn. 32 (1972), 332[JPSJ].
  39. F. Yoshida, T. Nakayama and T. Sakuma, J. Phys. Soc. Jpn. 40 (1976), 901[JPSJ].
  40. K. Miura, Thesis, Department of Computer Science, University of Illinois (1973).
  41. W. Sutherland, Lecture at the NSF Conference on the Theory and Application of Solitons, Tucson 1976.
  42. T. Shirafuji, Prog. Theor. Phys. Suppl. No. 59 (1976), 126[PTP].
  43. W. Sutherland, Lecture at the Conference on the Theory and Application of Solitons, Tucson, 1976 (to be published in Rocky Mountain Math. J.).
  44. H. Takahashi, Proc. Phys.-Math. Soc. Jpn. 24 (1942), 60.
    E. H. Lieb and D. C. Matis, Mathematical Physics in One Dimension (Academic Press, 1966), p. 25.

Citing Article(s) :

  1. Journal of the Physical Society of Japan 56 (1987) pp. 3491-3498 :
    The 3+1 Dimensional Toda Equation and its Exact Solutions
    Akira Nakamura
  2. Journal of the Physical Society of Japan 57 (1988) pp. 1594-1600 :
    Exact Solutions of the 2+1 Dimensional Toda Equation under Periodic Boundary Condition
    Akira Nakamura
  3. Journal of the Physical Society of Japan 59 (1990) pp. 4272-4278 :
    Explicit One-Soliton Solutions of the 1+1 Dimensional Toda Molecule Equation and Their Large Molecule Limit of the Infinite Toda Lattice
    Akira Nakamura
  4. Progress of Theoretical Physics Vol. 58 No. 3 (1977) pp. 1074-1076 :
    Scattering of a Soliton by an Impurity Atom in the Toda Lattice and Localized Modes
    Akira Nakamura and Shozo Takeno
  5. Progress of Theoretical Physics Vol. 58 No. 4 (1977) pp. 1114-1126 :
    Reflection and Transmission of Lattice Solitons
    Nobuo Yajima
  6. Progress of Theoretical Physics Vol. 60 No. 2 (1978) pp. 338-352 :
    Scattering of Lattice Solitons and the Excitation of Impurity Modes
    Fumio Yoshida and Tetsuro Sakuma
  7. Progress of Theoretical Physics Vol. 61 No. 2 (1979) pp. 427-442 :
    Surface Impurity Localized Mode Vibration of the Toda Lattice
    Akira Nakamura
  8. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 33-36 :
    Undamping of Strongly Nonlinear Surface Impurity Localized Mode in the Toda Lattice
    Akira Nakamura and Shozo Takeno
  9. Progress of Theoretical Physics Vol. 68 No. 6 (1982) pp. 1854-1863 :
    Reappearance of Ordered Motion in Non-Integrable Hamiltonian Systems
    M. K. Ali and R. L. Somorjai
  10. Progress of Theoretical Physics Vol. 71 No. 3 (1984) pp. 487-498 :
    Quasi-Solitons in Anharmonic Chains with Higher Neighbor Interactions and Long-Range Forces
    L. Solheim and M. K. Ali
  11. Progress of Theoretical Physics Vol. 96 No. 1 (1996) pp. 29-36 :
    On a Class of Rational and Mixed Soliton-Rational Solutions of the Toda Lattice
    A.S. Cârstea and D. Grecu
  12. Progress of Theoretical Physics Supplement No.59 (1976) pp. 148-161 :
    Chopping Phenomenon of a Nonlinear System
    Morikazu Toda, Ryogo Hirota and Junkichi Satsuma
  13. Progress of Theoretical Physics Supplement No.94 (1988) pp. 195-209 :
    Solitons in Higher Dimensions
    Akira Nakamura

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.